Despite being a relatively minor character in terms of ink spilled, Eowyn's character is very rich and developed.  Here's my take.  

On the topic of Eowyn's battle with the Witch King (and other scenes), see this video on how the movies missed a great opportunity: https://www.youtube.com/watch?v=MlQLmwRv5mU  

Intro music (The King of the Golden Hall) and outro music (Helm's Deep) from the Lord of the Rings Trilogy Soundtrack, copyright Reprise Records and New Line Productions.